Problem: Evaluate $~~\int^5_0 xe^{-x}dx\,$. Choose 1 answer: Choose 1 answer: (Choice A) A $-6e^{-5}+1$ (Choice B) B $-4e^{-5}+1$ (Choice C) C $-5e^{-5}-1$ (Choice D) D $-5e^{-5}+1$
Answer: We will solve this by integrating by parts. We know that $ \int u(x)v\,^\prime(x)dx = u(x)v(x)-\int u\,^\prime(x)v(x)dx\,$. We can rewrite this as $ \int u\ dv = uv-\int v\ du\,$. In this problem we will let $~u = x~$ and $~dv=e^{-x} dx\,$. Then $~du = dx~$ and $~v = \int e^{-x}dx = -e^{-x}$. Integration by parts gives $ \int^5_0 xe^{-x}dx =x\cdot\big(-e^{-x}\big)\Bigg]_0^5-\int^5_0\big(-e^{-x}\big)dx$ $ ~~~= \Big(-xe^{-x}-e^{-x}\Big)\Bigg]^5_0 $ $ ~~~= -e^{-x}(x+1)\Bigg]^5_0$ $ ~~~=-6e^{-5}-(-1\cdot1)=-6e^{-5}+1$